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Factorial moments

Factorial moments may be used to search for intermittency in events [Bia86]. The whole field was much studied in the late eighties and early nineties, and a host of different measures have been proposed. We only implement one of the original prescriptions.

To calculate the factorial moments, the full rapidity (or pseudorapidity) and azimuthal ranges are subdivided into bins of successively smaller size, and the multiplicity distributions in bins is studied. The program calculates pseudorapidity with respect to the $z$ axis; if desired, one could first find an event axis, e.g. the sphericity or thrust axis, and subsequently rotate the event to align this axis with the $z$ direction.

The full rapidity range $\vert y\vert < y_{\mathrm{max}}$ (or pseudorapidity range $\vert\eta\vert < \eta_{\mathrm{max}}$) and azimuthal range $0 < \varphi < 2\pi$ are subdivided into $m_y$ and $m_{\varphi}$ equally large bins. In fact, the whole analysis is performed thrice: once with $m_{\varphi}=1$ and the $y$ (or $\eta$) range gradually divided into 1, 2, 4, 8, 16, 32, 64, 128, 256 and 512 bins, once with $m_y = 1$ and the $\varphi$ range subdivided as above, and finally once with $m_y = m_{\varphi}$ according to the same binary sequence. Given the multiplicity $n_j$ in bin $j$, the $i$:th factorial moment is defined by

\begin{displaymath}
F_i = (m_y m_{\varphi})^{i-1} \, \sum_j
\frac{n_j(n_j-1)\cdots(n_j-i+1)}{n(n-1)\cdots(n-i+1)} ~.
\end{displaymath} (300)

Here $n = \sum_j n_j$ is the total multiplicity of the event within the allowed $y$ (or $\eta$) limits. The calculation is performed for the second through the fifth moments, i.e. $F_2$ through $F_5$.

The $F_i$ as given here are defined for the individual event, and have to be averaged over many events to give a reasonably smooth behaviour. If particle production is uniform and uncorrelated according to Poisson statistics, one expects $\langle F_i \rangle \equiv 1$ for all moments and all bin sizes. If, on the other hand, particles are locally clustered, factorial moments should increase when bins are made smaller, down to the characteristic dimensions of the clustering.


next up previous contents
Next: Routines and Common-Block Variables Up: Event Statistics Previous: Energy-Energy Correlation   Contents
Stephen_Mrenna 2012-10-24